Asymmetric propagation of acoustic waves is theoretically reported in a chirped phononic crystal made of the combination of two different nonlinear solids. The dispersion of the system is spatially dependent and allows the rainbow trapping inside the structure. Nonlinearity is used to activate the self-demodulation effect, which is enhanced due to the particular dispersion characteristics of the system. The performed numerical study reveals an efficient generation of the demodulated wave, up to 15% in terms of the pressure amplitude, as well as strong attenuation for undesired frequency components above the cut-off frequency. The obtained energy rectification ratio is in the order of 104 for the whole range of amplitudes employed in this work, indicating the robustness of the asymmetry and non-reciprocity of the proposed device for a wide operational range.